implies that d \alpha is arc-sine. (*) is not only sufficient but also necessary if d \alpha = wdx is absolutely continuous and either it has compact support or w(x) = \exp {-2Q(|x|)}(-oo < x < oo) where Q(x)(x \geq 0) is a positive increasing differentiable function for which x\rho Q'(x) is increasing for some \rho < 1. An example is constructed of an absolutely continuous arc-sine measure d \alpha for which (*) does not hold.
Following J.L.Ullman, loc. cit. we say that A \subset [-1,1] is a determining set if every absolutely continuous d \alpha = w(x)dx which satisfies A \subseteq {x: w(x) > 0 } \subseteq [-1,1] is arc-sine on [-1,1]. We give a proof of the conjecture of P.Erdös that a measurable set A is a determining set if and only if it has minimal logarithmic capacity 1/2.
Classif.: * 33A65 33A65 42C05 General theory of orthogonal functions and polynomials